Divide the following complex numbers. $ \dfrac{11+24i}{-4-i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-4+i}$ $ \dfrac{11+24i}{-4-i} = \dfrac{11+24i}{-4-i} \cdot \dfrac{{-4+i}}{{-4+i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(11+24i) \cdot (-4+i)} {(-4-i) \cdot (-4+i)} = \dfrac{(11+24i) \cdot (-4+i)} {(-4)^2 - (-1i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(11+24i) \cdot (-4+i)} {(-4)^2 - (-1i)^2} = $ $ \dfrac{(11+24i) \cdot (-4+i)} {16 + 1} = $ $ \dfrac{(11+24i) \cdot (-4+i)} {17} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({11+24i}) \cdot ({-4+i})} {17} = $ $ \dfrac{{11} \cdot {(-4)} + {24} \cdot {(-4) i} + {11} \cdot {1 i} + {24} \cdot {1 i^2}} {17} $ Evaluate each product of two numbers. $ \dfrac{-44 - 96i + 11i + 24 i^2} {17} $ Finally, simplify the fraction. $ \dfrac{-44 - 96i + 11i - 24} {17} = \dfrac{-68 - 85i} {17} = -4-5i $